{"id":"f1d074e7-3945-4845-9876-ede92d2eb620","arxiv_id":"2608.05522","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A k-nearest-neighbor forecast-error method with period-synchronized horizons and transient-slope consensus estimates negative Lyapunov exponents from short trajectory ensembles, recovering 92 of 112 logistic-map exponents and achieving R^2 around 0.98 on a no-fixed-point 2D map.","lead":"The paper introduces a data-driven way to measure how quickly a system settles back to stable behavior from short repeated recordings, without knowing the underlying equations. It reports accurate recovery on two benchmark maps, and argues the method could be applied to sensor and structural recovery data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4) is asserted: without a sensitivity check over K, N, and Y, the reported slopes could be finite-sample kNN artifacts rather than λ1.","rationale":"The reader's weakest-assumption analysis identifies Eq. (4) as the load-bearing premise, and I agree. My stress-test sharpens that concern: the missing derivation matters because the kNN forecast error in a noiseless deterministic system is fundamentally a finite-sample approximation error, not a direct measurement of infinitesimal perturbation growth. The empirical benchmarks are real and supportive, and the authors acknowledge important limitations (noise, continuous time, missing comparisons), but neither the logistic nor the 2D-map results isolate the mechanism. Varying K and N would settle whether the fitted slope is invariant to the estimator's finite-sample parameters; invariance is necessary for the slope to be interpretable as λ1. Because this check has not been performed, the current evidence is consistent with the claim but does not establish it. The reader's conditional verdict already reflects this uncertainty, so I do not change the verdict. I credit the paper for the transparent fixed-point example, the period-synchronization idea, the consensus rule, and the honest limitation statements, but the central equation remains an unverified proportionality rather than a demonstrated identity.","tokens_in":10296,"tokens_out":7329,"duration_ms":83542,"concrete_test":"Run the published pipeline on the logistic map at r=2.7 (known λ1=log 0.7) and at a stable period-2 parameter (e.g., r=3.2, λ1=(1/2)log|f'(x1)f'(x2)|), with N ∈ {500, 2000, 5000, 20000}, K ∈ {1,3,5,11,21}, Y=1 and Y=5, H=5, and all other hyperparameters unchanged. Plot the fitted slope versus log N and versus K. If the slope drifts by more than the per-fit uncertainty or deviates from λ1 by more than the reported MAE, Eq. (4) fails as a Lyapunov estimator. A second arm of the same test applies the pipeline to independent identically distributed scalar noise to characterize the null behavior of the consensus rule.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on Eq. (4): log E(h) ≈ C + λ1 h, with λ1 the dominant negative Lyapunov exponent. This is asserted in Sec. 2.1, not derived, and it is not a trivial consequence of contraction. In a deterministic noiseless stable system, the kNN forecast error is a finite-sample quantity: as N→∞ the kNN predictor becomes consistent and E(h)→0 for every stable system regardless of λ1. For finite N, the geometric-mean error is controlled by nearest-neighbor gaps, local density, K, embedding length Y, the train/test split, and the numerical floor, not only by the linearized contraction. The fixed-point example at r=2.7 is exact only because the KNN error there reduces to neighbor-gap contraction; it does not validate the general premise. The transient-consensus and acceptance rules in Sec. 2.3 select exactly the parameter points where a linear profile exists, so the reported R2/MAE values on accepted points cannot distinguish a true Lyapunov exponential from an artifact of the KNN geometry. The paper provides no derivation of Eq. (4), no sensitivity scan over K, N, or Y, and no control system with known λ1 but different KNN geometry. This is the load-bearing gap: if the slope changes with K or N, the estimator is not measuring λ1 even when the profile is linear and passes consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an equation-free estimator of the dominant negative largest Lyapunov exponent (LLE) from ensembles of short scalar trajectories. A k-nearest-neighbor predictor is trained on trajectory histories, the geometric-mean absolute forecast error is evaluated at period-synchronized horizons, and the slope of its logarithm is taken to estimate a negative LLE. A transient-slope consensus rule is used to accept or reject candidate estimates. The method is validated on the logistic map and on a two-dimensional no-fixed-point map, with reference exponents computed independently from the governing equations. The central claim is that, in a locally contracting regime, the log-error profile is approximately C + λ1 h, so the fitted slope estimates the dominant negative exponent without reconstructing a Jacobian or a surrogate model.","tokens_in":10489,"tokens_out":5234,"duration_ms":49407,"significance":"If the central premise holds, the method would be a practically useful tool for extracting contraction rates from repeated relaxation measurements, complementing positive-LLE estimators and avoiding the need for explicit tangent-space reconstruction. The paper is honest about limitations, and it explicitly distinguishes the target quantity as a 'dominant observable contraction rate' that may equal a negative LLE in some settings. The estimation stage does not use reference exponents during fitting or selection, and the benchmark systems have known reference values, which is a credit. However, the central relation in Eq. (4) is asserted rather than derived, and the reported accuracy metrics are computed only on accepted parameter points. The lack of sensitivity analysis and of all-point metrics means the headline numbers may overstate the method's reliability; the manuscript needs strengthening on these points before the claim is fully supported.","major_comments":[{"comment":"The central relation log E(h) ≈ C + λ1 h is asserted, not derived. It does not follow trivially from local contraction: for a deterministic noiseless stable system, the kNN predictor is consistent as N→∞ and E(h)→0 for any stable system regardless of λ1, while for finite N the error is controlled by neighbor-gap statistics, K, history length Y, embedding delay, and the numerical floor. The paper should provide either an analytic derivation under explicit assumptions (e.g., large N, low-dimensional observable, phase-aligned trajectories) or a sensitivity study showing that the fitted slope is invariant under variation of K, N, and Y. All reported results depend on Eq. (4), so this gap is load-bearing.","section":"Sec. 2.1, Eq. (4)"},{"comment":"The headline metrics (MAE, RMSE, R²) are computed only on accepted parameter points, after applying the consensus and acceptance rules. For the logistic negative branch, 92 of 112 reference points are accepted (82.14% coverage), so the reported MAE of 0.0253 and R² of 0.886 do not characterize the whole negative-exponent region. Moreover, the text says six isolated single-transient fits were rejected but the retained set contains 92 points; 112 − 6 = 106, so the fate of the remaining 14 points is unclear. Please report coverage-weighted or all-point metrics and clarify the missing-point accounting.","section":"Sec. 3.2, Table 1"},{"comment":"The acceptance criteria are not specified precisely: 'sufficient linearity', 'predominantly decreasing log-error values', 'limited contamination by exact-zero or numerical-floor errors', and the period-selection rule 'close to the minimum' in Eq. (5) are all qualitative. Because these rules determine which points enter the reported metrics, the paper should give exact thresholds and test the sensitivity of the results to those thresholds. Without this, the coverage and accuracy numbers are not reproducible.","section":"Sec. 2.3"},{"comment":"The fixed-point example at r = 2.7 is exact because, in one dimension, the kNN forecast error reduces to neighbor-gap contraction; it therefore does not validate the general higher-dimensional scalar-observable premise. The two-dimensional no-fixed-point benchmark is a more appropriate test, but the reported agreement across x_n, y_n, and the radial observable is not accompanied by any uncertainty quantification. A bootstrap or repeated-split analysis showing the distribution of estimated exponents would strengthen the claim that the slope is robust and not an artifact of a particular train/test partition.","section":"Sec. 3.1"}],"minor_comments":[{"comment":"The floor ϵ in the geometric-mean error is never defined; please state its value and how it is chosen, because it directly affects the log-error profile for strongly contracting systems.","section":"Eq. (3)"},{"comment":"The period-detection rule 'smallest period close to the minimum of Q_p' and the phrase 'confirmed in a shifted window' are not formalized; a precise criterion is needed for reproducibility.","section":"Sec. 2.2, Eq. (5)"},{"comment":"The positive-branch context row in Table 1 is described as a prior-work context calculation, but it is given the same prominence as the new negative-LLE results; please state explicitly that this row is not a new contribution of the present method.","section":"Sec. 3.2"},{"comment":"The paper nicely qualifies the target as a 'dominant observable contraction rate', but the title and abstract repeatedly say 'negative largest Lyapunov exponent'; please harmonize the terminology so that readers do not over-interpret the estimator as a full Lyapunov-spectrum method.","section":"Discussion"},{"comment":"Several references are incomplete (e.g., [22], [24], [34] lack complete volume/page information); please check the journal's reference style.","section":"References"},{"comment":"Figure 2(b) should distinguish accepted points from rejected or missing points with different markers, so that coverage is visually transparent rather than only summarized in Table 1.","section":"Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a genuine gap in data-driven stability estimation, and the benchmark design is reasonable. The central issue is that Eq. (4) is the entire foundation of the method and it is asserted rather than justified; a revision that adds either a derivation or a convincing sensitivity/ablation study would substantially raise the paper's value. The missing accounting of rejected points in Sec. 3.2 should also be corrected. The self-citation to the authors' prior positive-LLE work is used appropriately as context and does not itself indicate circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper gives a genuinely new estimator for negative Lyapunov exponents from short repeated trajectories, relies on an asserted contraction-slope assumption that is not derived, and validates on two maps with honest acknowledgment of the gaps. Send it to referees; the sensitivity analysis and a baseline comparison would decide whether the method is real or a kNN artifact.\n\nWhat's new: the combination of period-synchronized forecast horizons and transient-consensus acceptance, applied to geometric-mean forecast errors, is not in the cited literature. The 2D no-fixed-point test across three independent scalar observables is a strong check: the fact that the slopes agree despite no shared coordinates suggests the estimate is tied to the dynamics, not to a particular embedding. The paper is also straightforward about what it does not do: noise, experimental data, continuous-time systems, and like-for-like comparison with local-mapping, Floquet, or Koopman estimators are all deferred.\n\nThe soft spots are real but proportionate. The central equation (4) is asserted: log E(h) ~ C + λ1 h is not a trivial consequence of contraction, and for finite K/N the geometric-mean error is affected by neighbor-gap statistics and embedding parameters as much as by the linearized rate. The stress-test point about possible kNN finite-sample artifacts is fair: without a sweep over K, N, and Y, the reported slopes could be a density effect. The fixed-point example being exact to 1e-7 is a nice illustration, not a validation of the general premise. Reported R2 and MAE are computed only on accepted parameter points; no error bars or sensitivity analysis; no code or data shipped. These are standard weaknesses of a method paper and are all addressable.\n\nMy verdict: the paper deserves a serious referee. The idea is plausible, the benchmarks are real, and the limitations are acknowledged rather than hidden. I would ask the referees to specifically demand (1) a derivation or at least a careful finite-sample argument for Eq. (4), (2) sensitivity scans over K, N, Y and the transient/consensus thresholds, and (3) at least one comparison with an existing negative-exponent estimator on the same data. If those come back clean, this becomes a useful tool. For my own work, I wouldn't cite it yet, but I would track the revision.","headline":"A plausible equation-free negative-LLE estimator built on an asserted contraction premise; deserves a rigorous referee, not a desk reject.","tokens_in":11105,"tokens_out":2113,"would_cite":false,"duration_ms":20189,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D25","37M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A k-nearest-neighbor forecast-error slope, evaluated at period-matched horizons, estimates the dominant negative Lyapunov exponent directly from short trajectory ensembles without equations or a Jacobian.","keywords":["negative Lyapunov exponent","forecast-error contraction","k-nearest-neighbor prediction","short trajectory ensembles","period synchronization","stable periodic dynamics","equation-free estimation","Lyapunov exponent estimation"],"falsifier":"Take the logistic map at a strongly contracting parameter, for example $r=3.52$, and run the estimator with increasing ensemble sizes $N=100, 500, 2000, 5000$. If the consensus slope converges to the reference $\\lambda_1=\\langle \\log|r(1-2x_n)|\\rangle$ as $N$ grows, the identification in Eq. (4) is supported; if the slope plateaus at a less negative value while the linear profile persists, the fitted slope is an artifact of finite-ensemble or numerical-floor effects rather than the Lyapunov exponent.","tokens_in":9964,"feed_emoji":"📉","tokens_out":9848,"duration_ms":78686,"temperature":0.7,"pith_summary":"The paper claims that contraction, not just divergence, can be measured directly from data: in a stable regime, the geometric-mean error of a k-nearest-neighbor predictor decays exponentially with forecast horizon, and the decay rate is the dominant negative Lyapunov exponent. The proposed estimator trains the predictor on ensembles of short scalar trajectories, evaluates errors at horizons synchronized with the detected orbit period, and fits the logarithmic error profile. It accepts a slope only when several transient lengths agree, which guards against mistaking early transient decay for the asymptotic contraction rate. On the logistic map it recovers 92 of 112 negative-exponent parameter values with mean absolute error 0.0253 and $R^2=0.886$; on a two-dimensional no-fixed-point map, three independently processed scalar observables give MAE 0.00879–0.01145 and $R^2=0.983$–$0.986$. The significance, if true, is that negative Lyapunov exponents become available in settings where only repeated recovery responses are observed and no equations or Jacobian are known.","feed_headline":"Slope of forecast errors recovers negative Lyapunov exponents","feed_subtitle":"Nearest-neighbor forecast-error decay gives contraction rates from short recovery responses, without a Jacobian.","key_machinery":"The central object is the forecast-error contraction profile $\\log E(h) \\simeq C + \\lambda_1 h$, where $E(h)$ is the geometric-mean absolute error of a k-nearest-neighbor predictor at horizon $h$. The mechanism has three parts: (i) an ensemble of short post-transient trajectories supplies many comparable initial conditions; (ii) horizons are evaluated at multiples of the detected period $p$ so that the same phase of the periodic orbit is compared, turning an oscillating multibranch profile into a single contraction envelope; and (iii) a stable-transient consensus rule keeps only slopes that agree across several transient lengths, rejecting isolated linear fits that may represent transient decay rather than the asymptotic exponent. This machinery turns prediction-error decay into a scalar Lyapunov-exponent estimate without tangent-space reconstruction.","core_discovery":"The central claim is that a dominant negative Lyapunov exponent can be estimated without reconstructing the governing equations or Jacobian, by fitting the geometric-mean forecast error of a kNN predictor. In a locally contracting regime, $\\log E(h) \\simeq C + \\lambda_1 h$, so the slope of the logarithmic error profile is the exponent. Two design choices make this work for stable orbits: forecast horizons are synchronized to the detected period, comparing the same phase of the orbit, and candidate slopes are accepted only if they form a consensus across several transient lengths. The paper demonstrates the claim on two benchmarks: the logistic map, where 92 of 112 negative-exponent reference values are recovered with MAE 0.0253 and $R^2=0.886$, and a two-dimensional no-fixed-point map, where independent scalar pipelines for $x_n$, $y_n$, and $\\sqrt{x_n^2+y_n^2}$ give MAE 0.00879–0.01145 and $R^2=0.983$–$0.986$.","pith_inferences":["Inference: if the method is applied to a continuous-time limit cycle, the fitted slope should be interpreted as a transverse Lyapunov or Floquet decay exponent, not the largest exponent, which is zero along the phase direction; the paper gestures at this distinction but leaves its implementation for future work.","Inference: the observed conservative bias toward zero on strongly contracting logistic windows suggests a general identifiability limit: contraction rates faster than the noise floor divided by the available horizon cannot be recovered by any forecast-error estimator, no matter how the transient is chosen.","Inference: a direct practical test would be to compare the forecast-error contraction rate with conventional logarithmic decrement or damping estimates on the same bridge ring-down records; agreement is not guaranteed because a Lyapunov exponent is an observation-dependent nonlinear return rate rather than a modal damping ratio."],"forward_implications":["Negative Lyapunov exponents become estimable from data sets too short for classical divergence-based methods, because the estimator reads contraction before the signal hits the numerical floor.","The same estimator can be applied to repeated relaxation records, such as structural ring-down or event-triggered sensor responses, without a mechanistic model.","The reliability decision is part of the output: parameter values that fail the transient-consensus test are rejected rather than reported as high-confidence fits.","Independent scalar observables of a higher-dimensional system yield nearly the same contraction rate, so the observable does not need to be a full state vector.","Because the period-aware horizon is a scalar analogue of one-period Floquet comparison, the method may estimate transverse or Floquet decay rates for limit cycles when implemented in a phase-aligned way."],"supporting_citations":[{"why":"Supplies the forecast-error principle and the earlier positive-LLE estimator that this paper extends to contraction.","marker":"[9]"},{"why":"Supplies the two-dimensional no-fixed-point map used as the second benchmark and its reference dynamics.","marker":"[10]"},{"why":"Provides the tangent-map and orthogonalization algorithm used to compute reference Lyapunov exponents for validation.","marker":"[1]"},{"why":"Defines the classical time-series approach to Lyapunov exponents that motivates direct scalar estimation.","marker":"[2]"},{"why":"Provides an earlier neighbor-averaging negative-exponent estimator with which this method is contrasted.","marker":"[11]"},{"why":"Provides the radial-basis-function surrogate-Jacobian baseline that this method avoids.","marker":"[13]"},{"why":"Supplies the empirical Floquet-multiplier estimator whose one-period comparison resembles the period-synchronized horizon.","marker":"[15]"}],"fun_headline_variants":["Forecast-error slope recovers negative Lyapunov exponents","Stable dynamics from short trajectories, no Jacobian required","Period-aware forecast errors estimate contraction rates","Equation-free Lyapunov estimation from short trajectory ensembles","Forecast errors reveal stability in short sensor responses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the shrinkage of forecast errors measured from repeated short trajectories equals the true contraction rate of nearby states, so the slope of the error profile is the dominant negative Lyapunov exponent.","fun_headline_variants_meta":{"raw":{"variants":["Forecast-error slope recovers negative Lyapunov exponents","Stable dynamics from short trajectories, no Jacobian required","Period-aware forecast errors estimate contraction rates","Equation-free Lyapunov estimation from short trajectory ensembles","Forecast errors reveal stability in short sensor responses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000267,"raw_usage":{"total_tokens":1676,"prompt_tokens":1068,"completion_tokens":608,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":684,"completion_tokens_details":{"reasoning_tokens":534}},"tokens_in":684,"tokens_out":608,"duration_ms":6519,"temperature":1.0,"reasoning_tokens":534,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:32:39.863847+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the logistic map at a strongly contracting parameter, for example $r=3.52$, and run the estimator with increasing ensemble sizes $N=100, 500, 2000, 5000$. If the consensus slope converges to the reference $\\lambda_1=\\langle \\log|r(1-2x_n)|\\rangle$ as $N$ grows, the identification in Eq. (4) is supported; if the slope plateaus at a less negative value while the linear profile persists, the fitted slope is an artifact of finite-ensemble or numerical-floor effects rather than the Lyapunov exponent.","supporting_citations":[{"cited_title":"Chaotic map with no fixed points: Entropy, implementation and control,","cited_arxiv_id":null,"evidence_quote":"Supplies the two-dimensional no-fixed-point map used as the second benchmark and its reference dynamics."},{"cited_title":"A robust method on estimation of Lyapunov exponents from a noisy time series,","cited_arxiv_id":null,"evidence_quote":"Provides an earlier neighbor-averaging negative-exponent estimator with which this method is contrasted."},{"cited_title":"A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability,","cited_arxiv_id":null,"evidence_quote":"Provides the radial-basis-function surrogate-Jacobian baseline that this method avoids."},{"cited_title":"Development of Floquet multiplier estimator to determine nonlinear oscillatory behavior in power system data measurement,","cited_arxiv_id":null,"evidence_quote":"Supplies the empirical Floquet-multiplier estimator whose one-period comparison resembles the period-synchronized horizon."}],"review_version":1}